Deformation of Solids: Stress, Strain and the Young Modulus

This guide covers Topic 6, Deformation of solids, in full — subtopics 6.1 Stress and strain and 6.2 Elastic and plastic behaviour — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is AS Level content, and the final AS topic in the mechanics sequence that began with Kinematics.

Before studying this

This resource assumes work done and energy from Work, Energy and Power, since elastic potential energy is calculated from the area under a force–extension graph, the same technique used for work done from a force–displacement graph.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — AS Level, Topic 6

Forces and deformations in this topic are assumed to act in one dimension only.

6.1 Stress and strain — deformation caused by tensile or compressive forces; the terms load, extension, compression and limit of proportionality; Hooke’s law; the spring constant k = F/x; defining and using stress, strain and the Young modulus; describing an experiment to determine the Young modulus of a metal wire.

6.2 Elastic and plastic behaviour — the terms elastic deformation, plastic deformation and elastic limit; understanding that the area under a force–extension graph represents work done; determining elastic potential energy from that area; recalling and using Eₚ = ½Fx = ½kx² for a material deformed within its limit of proportionality.

Hooke’s law and the spring constant

Hooke’s law states that the extension of a material is proportional to the applied force, up to the limit of proportionality:

F = kx

where k is the spring constant, F is the load and x is the extension. Once the limit of proportionality is exceeded, extension is no longer proportional to force, even if the material is still behaving elastically.

Determining the Young modulus experimentally. A typical method loads a long, thin wire with increasing known weights, measuring extension with a vernier scale or travelling microscope against an unstretched reference wire (to cancel out effects like temperature expansion), then plots load against extension to find the gradient within the region of proportionality.

Stress, strain and the Young modulus

Stress = force / cross-sectional area (unit: Pa, equivalent to N m⁻²). Strain = extension / original length (a ratio, no unit). The Young modulus E is the ratio of stress to strain, valid only while the material obeys Hooke’s law:

E = stress / strain = (F/A) / (x/L)

Worked example. A wire of original length 2.0 m and cross-sectional area 1.0 × 10⁻⁶ m² extends by 1.5 mm under a load of 60 N.

stress = F/A = 60 / (1.0 × 10⁻⁶) = 6.0 × 10⁷ Pa
strain = x/L = (1.5 × 10⁻³) / 2.0 = 7.5 × 10⁻⁴
E = stress/strain = (6.0 × 10⁷) / (7.5 × 10⁻⁴) = 8.0 × 10¹⁰ Pa

Elastic and plastic deformation

Elastic deformation means the material returns to its original shape once the deforming force is removed. Plastic deformation means it does not — a permanent change in shape remains. The elastic limit is the point beyond which any further deformation becomes at least partly plastic; for many materials this is close to, but not identical with, the limit of proportionality.

Elastic potential energy

The area under a force–extension graph represents the work done in deforming the material. Where the material is deformed within its limit of proportionality (so F = kx holds), this area is a triangle, giving the elastic potential energy stored:

Eₚ = ½Fx = ½kx²

Comparing materials

Materials fall into three broad behavioural categories:

Type	Behaviour	Example
Brittle	Breaks at the elastic limit with little or no plastic deformation	Glass, ceramics
Ductile	Undergoes large plastic deformation before breaking; can be drawn into a wire	Copper
Polymeric	Capable of very large extensions; the loading and unloading curves differ	Rubber

Three separate properties are often confused: strong means a high breaking stress; stiff means a high Young modulus; tough means the material absorbs a large amount of energy before it fractures. These do not always go together — glass is both stiff and strong, since it resists stretching and withstands a high stress before breaking, but it is not tough, because it shatters with almost no plastic deformation to absorb energy first. Distinguishing strength, stiffness and toughness, rather than treating them as one property, is what separates a full-mark answer from a partial one.

Common mistakes

• Treating the limit of proportionality and the elastic limit as identical. They are related but distinct — proportionality can end slightly before the elastic limit is reached for some materials.
• Using Eₚ = ½Fx = ½kx² beyond the limit of proportionality. This formula only holds while F = kx is valid; beyond that, the force–extension graph is no longer a straight line and the work done must be found from the actual area under the loading curve, not this shortcut. That area only equals the recoverable elastic energy while the material is still behaving elastically (fully returns to its original length on unloading) – once any plastic deformation has occurred, some of that work is not recoverable, and the true recoverable energy is the smaller area under the separate unloading curve instead.
• Confusing stress (force per unit area) with pressure conceptually — they share units, but stress specifically describes an applied force producing deformation in a solid.
• Forgetting that strain has no unit, since it is a ratio of two lengths.

Quick revision checklist

• Hooke’s law, F = kx, and the limit of proportionality
• Definitions of stress, strain, and the Young modulus E = stress/strain
• An experimental method to determine the Young modulus of a wire
• Elastic vs. plastic deformation, and the elastic limit
• Elastic potential energy from the area under a force–extension graph, Eₚ = ½Fx = ½kx²

Related resources

• Work, Energy and Power — the work-done concept this topic’s energy formula builds on
• Physical Quantities, Units and Measurement — combining measurement uncertainties, relevant to the Young modulus experiment
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.